Find the slope of the tangent to the curve

Question

(a) Find the slope of the tangent to the curvey = 3 +4x^2 - 2x^3

at the point where .x = a

(b) Find equations of the tangent lines at the points (1,5)

and .(2,3)

; (c) Graph the curve and both tangents on a common screen.

Answer 1

Step 1 :

Definition:

The tangent line to the curve at the point is the line through point P with

slope    provided that this limit exists.

Step 2 :

(a)

The function is .

Substitute limit .

  

Step 3 :

(b)

The tangent line at point :  

Substitute in m :

Point-slope form .

Substitute  , and  .  

    

Step 4 :

The tangent line at point :

Substitute in m :

Point-slope form . 

Substitute  , and .

  

Step 5 :

c) The graph of the function and the tangent lines:

Solution :   

(a) At the point equation of tangent line is .

(b) At the point equation of tangent line is .

(c) The graph of the function and the tangent lines:

 

Answer 2

Step 1 :

Definition:

The tangent line to the curve at the point is the line through point P with

slope    provided that this limit exists.

Step 2 :

(a)

The function is .

Substitute limit .

  

Step 3 :

(b)

The tangent line at point :  

Substitute in m :

Point-slope form .

Substitute  , and  .  

    

Step 4 :

The tangent line at point :

Substitute in m :

Point-slope form . 

Substitute  , and .

  

Step 5 :

c) The graph of the function and the tangent lines:

Solution :   

(a) At the point equation of tangent line is .

(b) At the point equation of tangent line is .

(c) The graph of the function and the tangent lines: